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A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping  $  f $
if the facts that  $  f $
is a continuous mapping of the closure  $  \overline{G}\; $
of a domain  $  G $
onto the closure  $  \overline{D}\; $
of a domain  $  D $
and  $  f $
is a homeomorphism of  $  \overline{G}\; \setminus  G $
onto  $  \overline{D}\; \setminus  D $
imply that  $  f $
is a topological mapping of  $  \overline{G}\; $
onto  $  \overline{D}\; $. 
Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. [[Boundary correspondence (under conformal mapping)|Boundary correspondence (under conformal mapping)]]).

If  $  G $
and  $  D $
are plane domains with Euclidean boundaries homeomorphic to a circle and  $  D $
is bounded, then the principle of boundary correspondence holds for analytic functions  $  f $
on  $  G $, 
i.e.  $  f $
is a conformal mapping of  $  G $
onto  $  D $. 
In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [[#References|[1]]]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,   B.V. Shabat,   "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,   "On differentiable mappings"  ''Dokl. Akad. Nauk SSSR'' , '''95''' :  5  (1954)  pp. 921–923  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.I. Pinchuk,   "Holomorphic equivalence of certain classes of domains in $\CC^n$"  ''Math. USSR-Sb.'' , '''111 (153)''' :  1  (180)  pp. 67–94; 159  ''Mat. Sb.'' , '''111 (153)''' :  1  (1980)  pp. 67–94; 159</TD></TR></table>